On some arithmetic properties of Siegel functions
For \(r=(r_1,r_2)\in\mathbb Q^2\backslash\mathbb Z^2\), let the Siegel function \(g_r\) be defined by \[ g_r(\tau)=-q_\tau^{B_2(r_1)/2}e^{\pi ir_2(r_1-1)}(1-q_z)\prod_{n=1}^\infty (1-q_\tau^nq_z)(1-q_\tau^nq_z^{-1}), \] where \(q_z=e^{2\pi iz}\) with \(z=r_1\tau+r_2\) and \(B_2(x)=x^2-x+1/6\). Then it is known that a suitable product of Siegel functions will be a modular function on \(\Gamma(N)\). (See [\textit{D. S. Kubert} and \textit{S. Lang}, Modular units. New York etc.: Springer-Verlag (1981; Zbl 0492.12002)] for the necessary and sufficient conditions.) In the paper under review, the authors study several problems related to Siegel functions. Firstly, they show that the field of modular functions on \(\Gamma(N)\) is generated by the elliptic \(j\)-function, \(g^{12N}_{(1/N,0)}\), and \(g^{12N}_{(0,1/N)}\). They also discuss the cases of \(\Gamma_1(N)\) and \(\Gamma_0(N)\). Secondly, the authors determine when a product of Siegel functions is integral over \(\mathbb Z[j]\). As an application, they show that the reciprocals of the values of Ramanujan's cubic continued fraction at imaginary quadratic numbers are all algebraic integers. Finally, the authors use special values of Siegel functions to explicitly construct ray class fields of imaginary quadratic number fields.
scientific article; zbMATH DE number 5671533
scientific article